Wednesday, November 23, 2016

Plain English Explanation of Scottish STV

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Scottish STV is a great method to use for electing a group of people. To help you and your voters understand how it works, we give a "plain English" explanation of how votes are counted with Scottish STV.

If you'd like to see additional details, you are welcome to review the statute passed by Scotland.

At a high level, the vote count takes place in rounds.  For the first round, you count the first place votes.  For subsequent rounds, you will either (1) transfer surplus votes from a candidate or (2) eliminate a candidate.  We'll assume that you've read our STV overview and understand what surplus votes are.

You may find it helpful to review some Scottish STV election results while you are reading this.

Quota or Threshold

All STV methods have a quota or winning threshold.  The quota is computed using the number of non-empty votes.  For example, at the time of this writing, this election had 2006 total votes, but 252 of them were empty, so the quota is computed based on the 1754 non-empty votes.

The quota is computed by (1) dividing the number of non-empty votes by the number of seats plus one, (2) adding one, and (3) discarding any fraction.

For example, with 1754 non-empty votes and 2 seats, we compute 1754/3 + 1 and discard any fraction, which gives a quota of 585.

First Place Votes

The first step is quite easy. All votes are assigned to their first choices.

Transfer Surplus vs. Eliminate

At each round after the first, we need to either (1) transfer surplus votes from one candidate  or (2) eliminate one candidate.  The decision is easy. If at least one candidate has surplus votes, then we transfer surplus votes.  If no candidates have surplus votes, then we eliminate one candidate.

Note that in each round we process exactly one candidate. If more than one candidate has surplus votes, then we select the candidate with the largest number of votes and transfer only that candidate's surplus votes.  Other candidates with surplus votes will be processed in subsequent rounds.  If no candidates have surplus votes, then one candidate with the fewest number of votes is eliminated.

Once a candidate has transferred surplus votes or been eliminated, that candidate can no longer receive any votes.  For example, suppose in round 2 that A has a surplus and those surplus votes are transferred.  Suppose in round 3, that candidate B is eliminated and some of B's votes rank A second. For B's votes that rank A second, those votes will pass over the second choice of A and transfer to the third choice.

Transfer Surplus Votes

When transferring surplus votes from a candidate, we are actually transferring ALL votes received by that candidate, but we transfer them at a fraction of their value. The idea is that a quota of votes remain with the candidate and the surplus votes get counted towards the next choices on the ballots.

For example, suppose candidate A has 671 votes and that the quota is 585.  A thus has 86 surplus votes (671 - 585 = 86) that need to be transferred.

Candidate A has 671 votes, so we transfer each of them 86/671 of their value.  If we transfer 671 votes at a value of 86/671, then the total "number" of votes transferred is 86.

Fractions are hard (for people and computers) so the transfer value of 86/671 needs to be converted to a number, and the result is 0.128166915....  Most fractions result in an infinitely long number so the Scottish rules truncate to 5 digits or 0.12816. Accordingly, each of A's votes gets transferred at a value of 0.12816 to the next choice on the ballot.

Each candidate receiving votes from A may now have a total number of votes that isn't a whole number.  For example, candidate B may have had 478 votes before the transfer, received 26.65728 votes (208 of A's votes at a value of 0.12816), and now have 504.65728 votes.

Note that candidate B now has some votes with a value of 1 (e.g., the votes where B was ranked first) and some votes with a value of 0.12816 (the votes received in the surplus transfer from A).

Eliminate Candidate and Transfer Votes

Eliminating a candidate is much simpler.  All votes get transferred to their next choices at the same value that they currently have.

Exhausted Votes

Looking at Scottish STV results, you'll see a bar for "Exhausted" votes.  An exhausted vote occurs when a ballot has no next choice.  This can happen when you transfer surplus votes or votes from an eliminated candidate.  For example, if a voter ranked candidate A first and didn't rank anyone else, then that vote would become exhausted when transferring surplus votes from A and 0.12816 would be added to the exhausted bar.


Ties can occur when you are selecting a surplus to transfer or when selecting a candidate to eliminate. For example, suppose both candidate A and candidate B have surplus votes and that they have the same number of votes.  We now need to pick either A or B to transfer surplus votes.

With Scottish STV, ties are broken by looking to previous rounds.  Suppose A and B are tied at round 3.  To break the tie, we look to round 2 and see which candidate had more votes.  If A had more votes than B at round 2, then we select A to transfer surplus votes at round 3.  If A and B were also tied at round 2, then we look to round 1.  If they are tied at round 1, then we make a random selection.

Ending the Election

The election is over when all the winners are determined.  If there are N positions being filled, then the election is over when N candidates have reached the quota or when only N+1 candidates remain (the winners being the N candidates with the highest numbers of votes).


  1. Rounding question:
    You state that "Most fractions result in an infinitely long number so the Scottish rules truncate to 5 digits or 0.12816. Accordingly, each of A's votes gets transferred at a value of 0.12816 to the next choice on the ballot."

    When truncating, why not round up if the truncated number is 5 or more? It may not matter much in practice, but I'm curious as to why the truncation example you give is not rounded up. Do the Scots round that way? Is there a theoretical reason not to round that up?

    1. Hi Rick, all the STV rules I've seen round down. I think you could round as you suggest, but then the total number of votes could exceed the number of people who voted. I think that would confuse people and create a perception of something going wrong. If you always round down, you get a few more votes in the exhausted pile which seems more acceptable.

  2. This is a horrible system to count. I used to use it to count an eleven member seat with about 1000 votes. What is not mentioned here is that when you transfer a later surplus you have to re-calulate the values of the lower value surplus votes as well. At the end of the count nearly all the ballot papers were in two piles with about 50 different values (most very tiny) is the best counting system. between stages you have to do some sums to work out what to do next .. it is really not that complicated and you only have to transfer the pile of votes which caused the surplus which is a great time saver this or a similar system is used in Ireland. I don't know hat they were thinking when they invented the Scottish system!

    1. Hi Keith, yes, if being able to hand count the ballots easily is important, then the ERS97 rules ( are a better choice. OpaVote supports ERS97 rules as well.

    2. Keith and All - the Scots did not invent the Weighted Inclusive Gregory Method (WIGM) of transferring surpluses. It was a development from the defective Inclusive Gregory Method (IGM) used for federal STV-PR elections in Australia - see Farrell and McAllister (2003) Australian Journal of Political Science 38: 479 - 91. As Jeff said, the WIGM version of STV is not intended for manual counting (though it can be done). WIGM STV-PR was introduced in Scotland only after it was decided that we should use electronic counting (scanning conventional STV ballot papers) - the first implementation of WIGM STV for public elections anywhere in the world.

      The "inclusive" versions of the Gregory Method of fractional transfers were introduced because the 'last parcel' provision of the Gregory Method was considered by some to be "unfair" to significant numbers of voters. Consider the following election (only relevant candidates reported): Quota 100. Stage 1 first preference votes: Candidate A 101; Candidate B 99; Candidate C 5. All of these ballot papers are transferable as they have several 'next available preferences' marked.

      At Stage 1, Candidate A is elected. At Stage 2 Candidate A's surplus of one vote is transferred using all 101 ballot papers - none of those votes go to B or C and no other candidate achieves a quota. [Transfer value: 1 / 101 = 0.0099.] At Stage 3, Candidate C is excluded (eliminated) when two of her votes transfer to B, who is thereby elected (99 + 2 = 101). Stage 4 is the transfer of B's surplus. Under the 'last parcel' provision of the Gregory Method, only the two ballot papers received from C (the 'last parcel') will be used to transfer the surplus. [Transfer value: 1 / 2 = 0.5.] Giving the 99 voters who gave first preferences to B no say in the Stage 4 transfer has been considered "unfair" compared to the involvement of all of the 101 voters (comparable number) who gave their first preferences to Candidate A.

      To overcome this "unfairness" the Australians invented the "Inclusive Gregory Method" in which all of the relevant candidate's ballot papers are used whenever a surplus is transferred and the 'transfer value' is calculated as an average over all the ballot papers, no matter what value the ballot papers had when the now-elected candidate received them. Some simple arithmetic will show that this method is defective because it has the effect of increasing the value of some ballot papers above one vote, with compensatory reductions in the value of other ballot papers. Thus the Inclusive Gregory Method does not meet the fundamental requirement of "one person, one vote" and so should never be used for any kind of election anywhere. Despite the repeated demands of the Proportional Representation Society of Australia and others, the Australian authorities will not listen and they continue to use this defect procedure for federal STV-PR elections.

      The defect in the Inclusive Gregory Method was removed in the Weighted Inclusive Gregory Method by weighting each ballot paper by its current value when a new transfer value had to be calculated for the transfer of a surplus, as described by David Farrell and Ian McAllister. In this way all ballot papers retain their "one vote" value at all stages of the count.

      For a Detailed Description of a count in accordance with the current Election Rules for STV elections in Scotland see: